Distances in random graphs with finite mean and infinite variance degrees


 Elmer Lloyd
 2 years ago
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1 E e c t r o n i c J o u r n a o f P r o b a b i i t y Vo , Paper no. 25, pages Journa URL Distances in rando graphs with finite ean and infinite variance degrees Reco van der Hofstad Gerard Hooghiestra and Ditri Znaensi Abstract In this paper we study typica distances in rando graphs with i.i.d. degrees of which the tai of the coon distribution function is reguary varying with exponent 1 τ. Depending on the vaue of the paraeter τ we can distinct three cases: i τ >, where the degrees have finite variance, ii τ 2,, where the degrees have infinite variance, but finite ean, and iii τ 1, 2, where the degrees have infinite ean. The distances between two randoy chosen nodes beonging to the sae connected coponent, for τ > and τ 1, 2, have been studied in previous pubications, and we survey these resuts here. When τ 2,, the graph distance centers around 2 og og N/ ogτ 2. We present a fu proof of this resut, and study the fuctuations around this asyptotic eans, by describing the asyptotic distribution. The resuts presented here iprove upon resuts of Reittu and Norros, who prove an upper bound ony. The rando graphs studied here can serve as odes for copex networs where degree power aws are observed; this is iustrated by coparing the typica distance in this ode to Internet data, where a degree power aw with exponent τ 2.2 is observed for the socaed Autonoous Systes AS graph. Key words: Branching processes, configuration ode, couping, graph distance. Departent of Matheatics and Coputer Science, Eindhoven University of Technoogy, P.O. Box 51, 5600 MB Eindhoven, The Netherands. Eai: Supported in part by Netherands Organization for Scientific Research NWO. Deft University of Technoogy, Eectrica Engineering, Matheatics and Coputer Science, P.O. Box 501, 2600 GA Deft, The Netherands. Eai: EURANDOM, P.O. Box 51, 5600 MB Eindhoven, The Netherands. Eai: 70
2 AMS 2000 Subect Cassification: Priary 05C80; Secondary: 05C12, 60J80. Subitted to EJP on February 27, 2007, fina version accepted Apri 10,
3 1 Introduction Copex networs are encountered in a wide variety of discipines. A rough cassification has been given by Newan 18 and consists of: i Technoogica networs, e.g. eectrica power grids and the Internet, ii Inforation networs, such as the Word Wide Web, iii Socia networs, ie coaboration networs and iv Bioogica networs ie neura networs and protein interaction networs. What any of the above exapes have in coon is that the typica distance between two nodes in these networs are sa, a phenoenon that is dubbed the saword phenoenon. A second ey phenoenon shared by any of those networs is their scaefree nature; eaning that these networs have socaed poweraw degree sequences, i.e., the nuber of nodes with degree fas of as an inverse power of. We refer to 1; 18; 25 and the references therein for a further introduction to copex networs and any ore exapes where the above two properties hod. A rando graph ode where both the above ey features are present is the configuration ode appied to an i.i.d. sequence of degrees with a poweraw degree distribution. In this ode we start by saping the degree sequence fro a power aw and subsequenty connect nodes with the saped degree purey at rando. This ode autoaticay satisfies the power aw degree sequence and it is therefore of interest to rigorousy derive the typica distances that occur. Together with two previous papers 10; 14, the current paper describes the rando fuctuations of the graph distance between two arbitrary nodes in the configuration ode, where the i.i.d. degrees foow a power aw of the for PD > = τ+1 L, where L denotes a sowy varying function and the exponent τ satisfies τ 1. To obtain a copete picture we incude a discussion and a heuristic proof of the resuts in 10 for τ 1,2, and those in 14 for τ >. However, the ain goa of this paper is the copete description, incuding a fu proof of the case where τ 2,. Apart fro the critica cases τ = 2 and τ =, which depend on the behavior of the sowy varying function L see 10, Section 4.2 when τ = 2, we have thus given a copete anaysis for a possibe vaues of τ 1. This section is organized as foows. In Section 1.1, we start by introducing the ode, in Section 1.2 we state our ain resuts. Section 1. is devoted to reated wor, and in Section 1.4, we describe soe siuations for a better understanding of our ain resuts. Finay, Section 1.5 describes the organization of the paper. 1.1 Mode definition Fix an integer N. Consider an i.i.d. sequence D 1,D 2,...,D N. We wi construct an undirected graph with N nodes where node has degree D. We assue that = N =1 D is even. If is odd, then we increase D N by 1. This singe change wi ae hardy any difference in what foows, and we wi ignore this effect. We wi ater specify the distribution of D 1. To construct the graph, we have N separate nodes and incident to node, we have D stubs or hafedges. A stubs need to be connected to buid the graph. The stubs are nubered in any given order fro 1 to. We start by connecting at rando the first stub with one of the 705
4 1 reaining stubs. Once paired, two stubs hafedges for a singe edge of the graph. Hence, a stub can be seen as the eft or the right haf of an edge. We continue the procedure of randoy choosing and pairing the stubs unti a stubs are connected. Unfortunatey, nodes having sefoops ay occur. However, sefoops are scarce when N, as shown in 5. The above ode is a variant of the configuration ode, which, given a degree sequence, is the rando graph with that given degree sequence. The degree sequence of a graph is the vector of which the th coordinate equas the proportion of nodes with degree. In our ode, by the aw of arge nubers, the degree sequence is cose to the probabiity ass function of the noda degree D of which D 1,...,D N are independent copies. The probabiity ass function and the distribution function of the noda degree aw are denoted by x PD 1 = = f, = 1,2,..., and Fx = f, 1.1 where x is the argest integer saer than or equa to x. We consider distributions of the for =1 1 Fx = x τ+1 Lx, 1.2 where τ > 1 and L is sowy varying at infinity. This eans that the rando variabes D obey a power aw, and the factor L is eant to generaize the ode. We assue the foowing ore specific conditions, spitting between the cases τ 1, 2, τ 2, and τ >. Assuption 1.1. i For τ 1, 2, we assue 1.2. ii For τ 2,, we assue that there exists γ 0, 1 and C > 0 such that x τ+1 Cog xγ 1 1 Fx x τ+1+cog xγ 1, for arge x. 1. iii For τ >, we assue that there exists a constant c > 0 such that and that ν > 1, where ν is given by 1 Fx cx τ+1, for a x 1, 1.4 ν = ED 1D 1 1]. 1.5 ED 1 ] Distributions satisfying 1.4 incude distributions which have a ighter tai than a power aw, and 1.4 is ony sighty stronger than assuing finite variance. The condition in 1. is sighty stronger than Main resuts We define the graph distance H N between the nodes 1 and 2 as the iniu nuber of edges that for a path fro 1 to 2. By convention, the distance equas if 1 and 2 are not connected. Observe that the distance between two randoy chosen nodes is equa in distribution to H N, 706
5 because the nodes are exchangeabe. In order to state the ain resut concerning H N, we define the centering constant { 2 og og N ogτ 2, for τ 2,, τ,n = 1.6 og ν N, for τ >. The paraeter τ,n describes the asyptotic growth of H N as N. A ore precise resut incuding the rando fuctuations around τ,n is foruated in the foowing theore. Theore 1.2 The fuctuations of the graph distance. When Assuption 1.1 hods, then i for τ 1,2, where p = p F 0,1. i PH N = 2 = 1 i PH N = = p, 1.7 N N ii for τ 2, or τ > there exist rando variabes R τ,a a 1,0], such that as N, P H N = τ,n + H N < = PR τ,an = + o1, 1.8 where a N = { og og N og og N ogτ 2 ogτ 2, for τ 2,, og ν N og ν N, for τ >. We see that for τ 1,2, the iit distribution exists and concentrates on the two points 2 and. For τ 2, or τ > the iit behavior is ore invoved. In these cases the iit distribution does not exist, caused by the fact that the correct centering constants, 2og og N/ ogτ 2, for τ 2, and og ν N, for τ >, are in genera not integer, whereas H N is with probabiity 1 concentrated on the integers. The above theore cais that for τ 2, or τ > and arge N, we have H N = τ,n + O p 1, with τ,n specified in 1.6 and where O p 1 is a rando contribution, which is tight on R. The specific for of this rando contribution is specified in Theore 1.5 beow. In Theore 1.2, we condition on H N <. In the course of the proof, here and in 14, we aso investigate the probabiity of this event, and prove that PH N < = q 2 + o1, 1.9 where q is the surviva probabiity of an appropriate branching process. Coroary 1. Convergence in distribution aong subsequences. For τ 2, or τ >, and when Assuption 1.1 is fufied, we have that, for, H N τ,n H N < 1.10 converges in distribution to R τ,a, aong subsequences N where a N converges to a. A siuation for τ 2, iustrating the wea convergence in Coroary 1. is discussed in Section
6 Coroary 1.4 Concentration of the hopcount. For τ 2, or τ >, and when Assuption 1.1 is fufied, we have that the rando variabes H N τ,n, given that H N <, for a tight sequence, i.e., i K i sup P H N τ,n K N H N < = We next describe the aws of the rando variabes R τ,a a 1,0]. For this, we need soe further notation fro branching processes. For τ > 2, we introduce a deayed branching process {Z } 1, where in the first generation the offspring distribution is chosen according to 1.1 and in the second and further generations the offspring is chosen in accordance to g given by g = + 1f +1, = 0,1,..., where µ = ED 1 ] µ When τ 2,, the branching process {Z } has infinite expectation. Under Assuption 1.1, it is proved in 8 that i τ n 2n ogz n 1 = Y, a.s., 1.1 where x y denotes the axiu of x and y. When τ >, the process {Z n /µν n 1 } n 1 is a nonnegative artingae and consequenty i n Z n = W, a.s µνn 1 The constant q appearing in 1.9 is the surviva probabiity of the branching process {Z } 1. We can identify the iit aws of R τ,a a 1,0] in ters of the iit rando variabes in 1.1 and 1.14 as foows: Theore 1.5 The iit aws. When Assuption 1.1 hods, then i for τ 2, and for a 1,0], PR τ,a > = P in τ 2 s Y 1 +τ 2 s c Y 2] τ 2 /2 +a Y 1 Y 2 > 0, 1.15 s Z where c = 1 if is even and zero otherwise, and Y 1,Y 2 are two independent copies of the iit rando variabe in 1.1. ii for τ > and for a 1,0], PR τ,a > = E exp{ κν a+ W 1 W 2 } W 1 W 2 > 0 ], 1.16 where W 1 and W 2 are two independent copies of the iit rando variabe W in 1.14 and where κ = µν 1 1. The above resuts prove that the scaing in these rando graphs is quite sensitive to the degree exponent τ. The scaing of the distance between pairs of nodes is proved for a τ 1, except for the critica cases τ = 2 and τ =. The resut for τ 1,2, and the case τ = 1, where P H N 2, are both proved in 10, the resut for τ > is proved in 14. In Section 2 we wi present heuristic proofs for a three cases, and in Section 4 a fu proof for the case where 708
7 τ 2,. Theores quantify the saword phenoenon for the configuration ode, and expicity divide the scaing of the graph distances into three distinct regies In Rears 4.2 and A.1.5 beow, we wi expain that our resuts aso appy to the usua configuration ode, where the nuber of nodes with a given degree is deterinistic, when we study the graph distance between two unifory chosen nodes, and the degree distribution satisfied certain conditions. For the precise conditions, see Rear A.1.5 beow. 1. Reated wor There are any papers on scaefree graphs and we refer to reviews such as the ones by Abert and Barabási 1, Newan 18 and the recent boo by Durrett 9 for an introduction; we refer to 2; ; 17 for an introduction to cassica rando graphs. Papers invoving distances for the case where the degree distribution F see 1.2, has exponent τ 2, are not so wide spread. In this discussion we wi focus on the case where τ 2,. For reated wor on distances for the cases τ 1,2 and τ > we refer to 10, Section 1.4 and 14, Section 1.4, respectivey. The ode investigated in this paper with τ 2, was first studied in 21, where it was shown that with probabiity converging to 1, H N is ess than τ,n 1 + o1. We iprove the resuts in 21 by deriving the asyptotic distribution of the rando fuctuations of the graph distance around τ,n. Note that these resuts are in contrast to 19, Section II.F, beow Equation 56, where it was suggested that if τ <, then an exponentia cutoff is necessary to ae the graph distance between an arbitrary pair of nodes wedefined. The probe of the ean graph distance between an arbitrary pair of nodes was aso studied nonrigorousy in 7, where aso the behavior when τ = and x Lx is the constant function, is incuded. In the atter case, og N ogog N the graph distance scaes ie. A reated ode to the one studied here can be found in 20, where a Poissonian graph process is defined by adding and reoving edges. In 20, the authors prove siiar resuts as in 21 for this reated ode. For τ 2,, in 15, it was further shown that the diaeter of the configuration ode is bounded beow by a constant ties og N, when f 1 + f 2 > 0, and bounded above by a constant ties og og N, when f 1 + f 2 = 0. A second reated ode can be found in 6, where edges between nodes i and are present with probabiity equa to w i w / w for soe expected degree vector w = w 1,...,w N. It is assued that ax i wi 2 < i w i, so that w i w / w are probabiities. In 6, w i is often taen as w i = ci 1 1 τ 1, where c is a function of N proportiona to N τ 1. In this case, the degrees obey a power aw with exponent τ. Chung and Lu 6 show that in this case, the graph distance between two unifory chosen nodes is with probabiity converging to 1 proportiona to og N1 + o1 og og N when τ >, and to 2 ogτ o1 when τ 2,. The difference between this ode and ours is that the nodes are not exchangeabe in 6, but the observed phenoena are siiar. This resut can be heuristicay understood as foows. Firsty, the actua degree vector in 6 shoud be cose to the expected degree vector. Secondy, for the expected degree vector, we can copute that the nuber of nodes for which the degree is at east equas {i : w i } = {i : ci 1 τ 1 } τ+1. Thus, one expects that the nuber of nodes with degree at east decreases as τ+1, siiary as in our ode. The ost genera version of this ode can be found in 4. A these odes 709
8 Figure 1: Histogras of the AScount and graph distance in the configuration ode with N = 10,940, where the degrees have generating function f τ s in 1.18, for which the power aw exponent τ taes the vaue τ = The ASdata is ighty shaded, the siuation is dary shaded. assue soe for of conditiona independence of the edges, which resuts in asyptotic degree sequences that are given by ixed Poisson distributions see e.g. 5. In the configuration ode, instead, the degrees are independent. 1.4 Deonstration of Coroary 1. Our otivation to study the above version of the configuration ode is to describe the topoogy of the Internet at a fixed tie instant. In a seina paper 12, Faoutsos et a. have shown that the degree distribution in Internet foows a power aw with exponent τ Thus, the power aw rando graph with this vaue of τ can possiby ead to a good Internet ode. In 24, and inspired by the observed power aw degree sequence in 12, the power aw rando graph is proposed as a ode for the networ of autonoous systes. In this graph, the nodes are the autonoous systes in the Internet, i.e., the parts of the Internet controed by a singe party such as a university, copany or provider, and the edges represent the physica connections between the different autonoous systes. The wor of Faoutsos et a. in 12 was aong others on this graph which at that tie had size approxiatey 10,000. In 24, it is argued on a quaitative basis that the power aw rando graph serves as a better ode for the Internet topoogy than the currenty used topoogy generators. Our resuts can be seen as a step towards the quantitative understanding of whether the AScount in Internet is described we by the graph distance in the configuration ode. The AScount gives the nuber of physica ins connecting the various autonoous doains between two randoy chosen doains. To vaidate the ode studied here, we copare a siuation of the distribution of the distance between pairs of nodes in the configuration ode with the sae vaue of N and τ to extensive easureents of the AScount in Internet. In Figure 1, we see that the graph distance in the ode with the predicted vaue of τ = 2.25 and the vaue of N fro the data set fits the AScount data rearaby we. 710
9 Figure 2: Epirica surviva functions of the graph distance for τ = 2.8 and for the four vaues of N. Having otivated why we are interested to study distances in the configuration ode, we now expain by a siuation the reevance of Theore 1.2 and Coroary 1. for τ 2,. We have chosen to siuate the distribution 1.12 using the generating function: τ 2 g τ s = 1 1 s τ 2, for which g = 1 1 c τ 1, Defining it is iediate that f τ s = τ sτ 1 s, τ 2,, 1.18 τ 2 τ 2 g τ s = f τ s f τ 1, so that g = + 1f +1. µ For fixed τ, we can pic different vaues of the size of the siuated graph, so that for each two siuated vaues N and M we have a N = a M, i.e., N = M τ 2, for soe integer. For τ = 2.8, this induces, starting fro M = 1000, by taing for the successive vaues 1,2,, M = 1,000, N 1 = 5,624, N 2 = 48,697, N = 72,95. According to Coroary 1., the surviva functions of the hopcount H N, given by PH N > H N <, and for N = M τ 2, run approxiatey parae on distance 2 in the iit for N, since τ,n = τ,m + 2 for = 1,2,. In Section.1 beow we wi show that the distribution with generating function 1.18 satisfies Assuption 1.1ii. 711
10 1.5 Organization of the paper The paper is organized as foows. In Section 2 we heuristicay expain our resuts for the three different cases. The reevant iterature on branching processes with infinite ean. is reviewed in Section, where we aso describe the growth of shortest path graphs, and state couping resuts needed to prove our ain resuts, Theores in Section 4. In Section 5, we prove three technica eas used in Section 4. We finay prove the couping resuts in the Appendix. In the seque we wi write that event E occurs whp for the stateent that PE = 1 o1, as the tota nuber of nodes N. 2 Heuristic expanations of Theores 1.2 and 1.5 In this section, we present a heuristic expanation of Theores 1.2 and 1.5. When τ 1,2, the tota degree is the i.i.d. su of N rando variabes D 1,D 2,...,D N, with infinite ean. Fro extree vaue theory, it is we nown that then the bu of the contribution to coes fro a finite nuber of nodes which have giant degrees the socaed giant nodes. Since these giant nodes have degree roughy N 1/τ 1, which is uch arger than N, they are a connected to each other, thus foring a copete graph of giant nodes. Each stub of node 1 or node 2 is with probabiity cose to 1 attached to a stub of soe giant node, and therefore, the distance between any two nodes is, whp, at ost. In fact, this distance equas 2 precisey when the two nodes are attached to the sae giant node, and is otherwise. For τ = 1 the quotient M N /, where M N denotes the axiu of D 1,D 2,...,D N, converges to 1 in probabiity, and consequenty the asyptotic distance is 2 in this case, as basicay a nodes are attached to the unique giant node. As entioned before, fu proofs of these resuts can be found in 10. For τ 2, or τ > there are two basic ingredients underying the graph distance resuts. The first one is that for two disoint sets of stubs of sizes n and out of a tota of L, the probabiity that none of the stubs in the first set is attached to a stub in the second set, is approxiatey equa to n 1 i= L n 2i In fact, the product in 2.1 is precisey equa to the probabiity that none of the n stubs in the first set of stubs is attached to a stub in the second set, given that no two stubs in the first set are attached to one another. When n = ol, L, however, these two probabiities are asyptoticay equa. We approxiate 2.1 further as n 1 i=0 1 { n 1 exp L n 2i i=0 og n + 2i } e n L, 2.2 L L where the approxiation is vaid as ong as nn + = ol 2, when L. The shortest path graph SPG fro node 1 is the union of a shortest paths between node 1 and a other nodes {2,...,N}. We define the SPG fro node 2 in a siiar fashion. We appy the above heuristic asyptotics to the growth of the SPG s. Let Z 1,N denote the nuber of stubs 712
11 that are attached to nodes precisey 1 steps away fro node 1, and siiary for Z 2,N. We then appy 2.2 to n = Z 1,N, = Z 2,N and L =. Let Q, Z be the conditiona distribution given {Z s 1,N } s=1 and {Z2,N s } s=1. For = 0, we ony condition on {Z1,N s } s=1. For 1, we have the utipication rue see 14, Lea 4.1, PH N > = E +1 i=2 ] Q i/2, i/2 Z H N > i 1 H N > i 2, 2. where x is the saest integer greater than or equa to x and x the argest integer saer than or equa to x. Now fro 2.1 and 2.2 we find, } Q i/2, i/2 Z H N > i 1 H N > i 2 exp { Z1,N i/2 Z2,N i/ This asyptotic identity foows because the event {H N > i 1 H N > i 2} occurs precisey when none of the stubs Z 1,N i/2 attaches to one of those of Z2,N i/2. Consequenty we can approxiate PH N > E exp { 1 +1 A typica vaue of the hopcount H N is the vaue for which i=2 Z 1,N i/2 Z2,N i/2 }] i=2 Z 1,N i/2 Z2,N i/2 1. This is the first ingredient of the heuristic. The second ingredient is the connection to branching processes. Given any node i and a stub attached to this node, we attach the stub to a second stub to create an edge of the graph. This chosen stub is attached to a certain node, and we wish to investigate how any further stubs this node has these stubs are caed brother stubs of the chosen stub. The conditiona probabiity that this nuber of brother stubs equas n given D 1,...,D N, is approxiatey equa to the probabiity that a rando stub fro a = D D N stubs is attached to a node with in tota n + 1 stubs. Since there are precisey N =1 n + 11 {D =n+1} stubs that beong to a node with degree n + 1, we find for the atter probabiity g N n = n + 1 N 1 {D =n+1}, 2.6 =1 where 1 A denotes the indicator function of the event A. The above forua coes fro saping with repaceent, whereas in the SPG the saping is perfored without repaceent. Now, as we grow the SPG s fro nodes 1 and 2, of course the nuber of stubs that can sti be chosen decreases. However, when the size of both SPG s is uch saer than N, for instance at ost N, or sighty bigger, this dependence can be negected, and it is as if we choose each tie independenty and with repaceent. Thus, the growth of the SPG s is cosey reated to a branching process with offspring distribution {g n N } n=1. 71
12 When τ > 2, using the strong aw of arge nubers for N, N µ = ED 1], and 1 N N 1 {D =n+1} f n+1 = PD 1 = n + 1, =1 so that, aost surey, g N n n + 1f n+1 µ = g n, N. 2.7 Therefore, the growth of the shortest path graph shoud be we described by a branching process with offspring distribution {g n }, and we coe to the question what is a typica vaue of for which +1 i=2 Z 1 i/2 Z2 i/2 = µn, 2.8 where {Z 1 } and {Z 2 } denote two independent copies of a deayed branching process with offspring distribution {f n }, f n = PD = n, n = 1,2,..., in the first generation and offspring distribution {g n } in a further generations. To answer this question, we need to ae separate arguents depending on the vaue of τ. When τ >, then ν = n 1 ng n <. Assue aso that ν > 1, so that the branching process is supercritica. In this case, the branching process Z /µν 1 converges aost surey to a rando variabe W see Hence, for the two independent branching processes {Z i }, i = 1,2, that ocay describe the nuber of stubs attached to nodes on distance 1, we find that, for, Z i µν 1 W i. 2.9 This expains why the average vaue of Z i,n grows ie µν 1 = µ exp 1og ν, that is, exponentia in for ν > 1, so that a typica vaue of for which 2.8 hods satisfies µ ν 1 = N, or = og ν N/µ + 1. We can extend this arguent to describe the fuctuation around the asyptotic ean. Since 2.9 describes the fuctuations of Z i around the ean vaue µν 1, we are abe to describe the rando fuctuations of H N around og ν N. The detais of these proofs can be found in 14. When τ 2,, the branching processes {Z 1 } and {Z 2 } are wedefined, but they have infinite ean. Under certain conditions on the underying offspring distribution, which are ipied by Assuption 1.1ii, Davies 8 proves for this case that τ 2 ogz +1 converges aost surey, as, to soe rando variabe Y. Moreover, PY = 0 = 1 q, the extinction probabiity of {Z } =0. Therefore, aso τ 2 ogz 1 converges aost surey to Y. Since τ > 2, we sti have that µn. Furtherore by the doube exponentia behavior of Z i, the size of the efthand side of 2.8 is equa to the size of the ast ter, so that the typica vaue of for which 2.8 hods satisfies Z 1 +1/2 Z2 +1/2 µn, or ogz1 +1/2 1 + ogz2 +1/2 1 og N. This indicates that the typica vaue of is of order og og N 2 ogτ 2,
13 as foruated in Theore 1.2ii, since if for soe c 0,1 ogz 1 +1/2 1 cog N, ogz2 +1/2 1 1 cog N then + 1/2 = ogcog N/ ogτ 2, which induces the eading order of τ,n defined in 1.6. Again we stress that, since Davies resut 8 describes a distributiona iit, we are abe to describe the rando fuctuations of H N around τ,n. The detais of the proof are given in Section 4. The growth of the shortest path graph In this section we describe the growth of the shortest path graph SPG. This growth reies heaviy on branching processes BP s. We therefore start in Section.1 with a short review of the theory of BP s in the case where the expected vaue ean of the offspring distribution is infinite. In Section.2, we discuss the couping between these BP s and the SPG, and in Section., we give the bounds on the couping. Throughout the reaining sections of the seque we wi assue that τ 2,, and that F satisfies Assuption 1.1ii..1 Review of branching processes with infinite ean In this review of BP s with infinite ean we foow in particuar 8, and aso refer the readers to reated wor in 22; 2, and the references therein. For the fora definition of the BP we define a doube sequence {X n,i } n 0,i 1 of i.i.d. rando variabes each with distribution equa to the offspring distribution {g } given in 1.12 with distribution function Gx = x =0 g. The BP {Z n } is now defined by Z 0 = 1 and Z n Z n+1 = X n,i, n 0. i=1 In case of a deayed BP, we et X 0,1 have probabiity ass function {f }, independenty of {X n,i } n 1. In this section we restrict to the nondeayed case for sipicity. We foow Davies in 8, who gives the foowing sufficient conditions for convergence of τ 2 n og1 + Z n. Davies ain theore states that if there exists a nonnegative, nonincreasing function γx, such that, i x ζ γx 1 Gx x ζ+γx, for arge x and 0 < ζ < 1, ii x γx is nondecreasing, iii 0 γe ex dx <, or, equivaenty, γy e y og y dy <, then ζ n og1 + Z n converges aost surey to a nondegenerate finite rando variabe Y with PY = 0 equa to the extinction probabiity of {Z n }, whereas Y Y > 0 adits a density on 0,. Therefore, aso ζ n ogz n 1 converges to Y aost surey. 715
14 The conditions of Davies quoted as iiii sipify earier wor by Seneta 2. For exape, for {g } in 1.17, the above is vaid with ζ = τ 2 and γx = Cog x 1, where C is sufficienty arge. We prove in Lea A.1.1 beow that for F as in Assuption 1.1ii, and G the distribution function of {g } in 1.12, the conditions iiii are satisfied with ζ = τ 2 and γx = Cog x γ 1, with γ < 1. Let Y 1 and Y 2 be two independent copies of the iit rando variabe Y. In the course of the proof of Theore 1.2, for τ 2,, we wi encounter the rando variabe U = in t Z κ t Y 1 + κ c t Y 2, for soe c {0,1}, and where κ = τ 2 1. The proof reies on the fact that, conditionay on Y 1 Y 2 > 0, U has a density. The proof of this fact is as foows. The function y 1,y 2 in t Z κ t y 1 + κ c t y 2 is discontinuous precisey in the points y 1,y 2 satisfying y 2 /y 1 = κ n 1 2 c, n Z, and, conditionay on Y 1 Y 2 > 0, the rando variabes Y 1 and Y 2 are independent continuous rando variabes. Therefore, conditionay on Y 1 Y 2 > 0, the rando variabe U = in t Z κ t Y 1 + κ c t Y 2 has a density..2 Couping of SPG to BP s In Section 2, it has been shown inforay that the growth of the SPG is cosey reated to a BP } with the rando offspring distribution {g N } given by 2.6; note that in the notation {Ẑ1,N Ẑ 1,N we do incude its dependence on N, whereas in 14, Section.1 this dependence on N was eft out for notationa convenience. The presentation in Section.2 is virtuay identica to the one in 14, Section. However, we have decided to incude ost of this ateria to eep the paper sefcontained. By the strong aw of arge nubers, g N + 1PD 1 = + 1/ED 1 ] = g, N. Therefore, the BP {Ẑ1,N }, with offspring distribution {g N }, is expected to be cose to the BP {Z 1 } with offspring distribution {g } given in So, in fact, the couping that we ae is twofod. We first coupe the SPG to the N dependent branching process {Ẑ1,N }, and consecutivey we coupe {Ẑ1,N } to the BP {Z 1 }. In Section., we state bounds on these coupings, which aow us to prove Theores 1.2 and 1.5 of Section 1.2. The shortest path graph SPG fro node 1 consists of the shortest paths between node 1 and a other nodes {2,...,N}. As wi be shown beow, the SPG is not necessariy a tree because cyces ay occur. Reca that two stubs together for an edge. We define Z 1,N 1 = D 1 and, for 2, we denote by Z 1,N the nuber of stubs attached to nodes at distance 1 fro node 1, but are not part of an edge connected to a node at distance 2. We refer to such stubs as free stubs, since they have not yet been assigned to a second stub to fro an edge. Thus, Z 1,N is the nuber of outgoing stubs fro nodes at distance 1 fro node 1. By SPG 1 we denote the SPG up to eve 1, i.e., up to the oent we have Z 1,N free stubs attached to nodes on distance 1, and no stubs to nodes on distance. Since we copare Z 1,N to the th generation of the BP Ẑ1,N, we ca Z 1,N the stubs of eve. For the copete description of the SPG {Z 1,N }, we have introduced the concept of abes in 14, Section. These abes iustrate the resebances and the differences between the SPG {Z 1,N } and the BP {Ẑ1,N }. 716
15 SPG stubs with their abes Figure : Scheatic drawing of the growth of the SPG fro the node 1 with N = 9 and the updating of the abes. The stubs without a abe are understood to have abe 1. The first ine shows the N different nodes with their attached stubs. Initiay, a stubs have abe 1. The growth process starts by choosing the first stub of node 1 whose stubs are abeed by 2 as iustrated in the second ine, whie a the other stubs aintain the abe 1. Next, we unifory choose a stub with abe 1 or 2. In the exape in ine, this is the second stub fro node, whose stubs are abeed by 2 and the second stub by abe. The eft hand side coun visuaizes the growth of the SPG by the attachent of stub 2 of node to the first stub of node 1. Once an edge is estabished the paired stubs are abeed. In the next step, again a stub is chosen unifory out of those with abe 1 or 2. In the exape in ine 4, it is the first stub of the ast node that wi be attached to the second stub of node 1, the next in sequence to be paired. The ast ine exhibits the resut of creating a cyce when the first stub of node is chosen to be attached to the second stub of node 9 the ast node. This process is continued unti there are no ore stubs with abes 1 or 2. In this exape, we have Z 1,N 1 = and Z 1,N 2 = 6. Initiay, a stubs are abeed 1. At each stage of the growth of the SPG, we draw unifory at rando fro a stubs with abes 1 and 2. After each draw we wi update the reaization of the SPG according to three categories, which wi be abeed 1, 2 and. At any stage of the generation of the SPG, the abes have the foowing eaning: 1. Stubs with abe 1 are stubs beonging to a node that is not yet attached to the SPG. 2. Stubs with abe 2 are attached to the SPG because the corresponding node has been chosen, but not yet paired with another stub. These are the free stubs entioned above.. Stubs with abe in the SPG are paired with another stub to for an edge in the SPG. 717
16 The growth process as depicted in Figure starts by abeing a stubs by 1. Then, because we construct the SPG starting fro node 1 we reabe the D 1 stubs of node 1 with the abe 2. We note that Z 1,N 1 is equa to the nuber of stubs connected to node 1, and thus Z 1,N 1 = D 1. We next identify Z 1,N for > 1. Z 1,N is obtained by sequentiay growing the SPG fro the free stubs in generation Z 1,N 1. When a free stubs in generation 1 have chosen their connecting stub, Z 1,N is equa to the nuber of stubs abeed 2 i.e., free stubs attached to the SPG. Note that not necessariy each stub of Z 1,N 1 contributes to stubs of Z1,N, because a cyce ay swaow two free stubs. This is the case when a stub with abe 2 is chosen. After the choice of each stub, we update the abes as foows: 1. If the chosen stub has abe 1, we connect the present stub to the chosen stub to for an edge and attach the brother stubs of the chosen stub as chidren. We update the abes as foows. The present and chosen stub et together to for an edge and both are assigned abe. A brother stubs receive abe When we choose a stub with abe 2, which is aready connected to the SPG, a sefoop is created if the chosen stub and present stub are brother stubs. If they are not brother stubs, then a cyce is fored. Neither a sefoop nor a cyce changes the distances to the root in the SPG. The updating of the abes soey consists of changing the abe of the present and the chosen stubs fro 2 to. The above process stops in the th generation when there are no ore free stubs in generation 1 for the SPG, and then Z 1,N is the nuber of free stubs at this tie. We continue the above process of drawing stubs unti there are no ore stubs having abe 1 or 2, so that a stubs have abe. Then, the SPG fro node 1 is finaized, and we have generated the shortest path graph as seen fro node 1. We have thus obtained the structure of the shortest path graph, and now how any nodes there are at a given distance fro node 1. The above construction wi be perfored identicay fro node 2, and we denote the nuber of free stubs in the SPG of node 2 in generation by Z 2,N. This construction is cose to being independent, when the generation size is not too arge. In particuar, it is possibe to coupe the two SPG growth processes with two independent BP s. This is described in detai in 14, Section. We ae essentia use of the couping between the SPG s and the BP s, in particuar, of 14, Proposition A..1 in the appendix. This copetes the construction of the SPG s fro both node 1 and 2.. Bounds on the couping We now investigate the reationship between the SPG {Z i,n } and the BP {Z i } with aw g. These resuts are stated in Proposition.1,.2 and.4. In their stateent, we write, for i = 1,2, Y i,n = τ 2 ogz i,n 1 and Y i = τ 2 ogz i 1,.1 where {Z 1 } 1 and {Z 2 } 1 are two independent deayed BP s with offspring distribution {g } and where Z i 1 has aw {f }. Then the foowing proposition shows that the first eves of the SPG are cose to those of the BP: 718
17 Proposition.1 Couping at fixed tie. If F satisfies Assuption 1.1ii, then for every fixed, and for i = 1,2, there exist independent deayed BP s Z 1, Z 2, such that i,n i PY N = Y i = 1..2 In words, Proposition.1 states that at any fixed tie, the SPG s fro 1 and 2 can be couped to two independent BP s with offspring g, in such a way that the probabiity that the SPG differs fro the BP vanishes when N. In the stateent of the next proposition, we write, for i = 1,2, T i,n = T i,n ε = { > : Z i,n κ 1 ε 2 N = { > : κ Y i,n 1 ε2 τ 1 τ 1 } og N},. where we reca that κ = τ 2 1. We wi see that Z i,n grows superexponentiay with as ong as T i,n. More precisey, Z i,n is cose to Z i,n κ, and thus, T i,n can be thought of as the generations for which the generation size is bounded by N 1 ε2 τ 1. The second ain resut of the couping is the foowing proposition: Proposition.2 Superexponentia growth with base Y i,n Assuption 1.1ii, then, for i = 1,2, a P ε Y i,n ε 1, ax T i,n ε Y i,n for arge ties. If F satisfies Y i,n > ε = o N,,ε1,.4 b P ε Y i,n P ε Y i,n ε 1, T i,n ε : Z i,n 1 > Zi,N ε 1, T i,n ε : Z i,n > N 1 ε 4 τ 1 = o N,,ε1,.5 = o N,,ε1,.6 where o N,,ε1 denotes a quantity γ N,,ε that converges to zero when first N, then and finay ε 0. Rear.. Throughout the paper iits wi be taen in the above order, i.e., first we send N, then and finay ε 0. Proposition.2 a, i.e..4, is the ain couping resut used in this paper, and says that as ong as T i,n ε, we have that Y i,n is cose to Y i,n, which, in turn, by Proposition.1, is cose to Y i. This estabishes the couping between the SPG and the BP. Part b is a technica resut used in the proof. Equation.5 is a convenient resut, as it shows that, with high probabiity, Z i,n is onotonicay increasing. Equation.6 shows that with high 4 for a T i,n ε. probabiity Z i,n N 1 ε τ 1 stubs in generation sizes that are in T i,n ε, which aows us to bound the nuber of free We copete this section with a fina couping resut, which shows that for the first which is not in T i,n ε, the SPG has any free stubs: 719
18 Proposition.4 Lower bound on Z i,n +1 Then, P T i,n ε, + 1 T i,n ε,ε Y i,n i,n for +1 T ε. Let F satisfy Assuption 1.1ii. ε 1,Z i,n +1 N 1 ε τ 1 = o N,,ε1..7 Propositions.1,.2 and.4 wi be proved in the appendix. In Section 4 and 5, we wi prove the ain resuts in Theores 1.2 and 1.5 subect to Propositions.1,.2 and.4. 4 Proof of Theores 1.2 and 1.5 for τ 2, For convenience we cobine Theore 1.2 and Theore 1.5, in the case that τ 2,, in a singe theore that we wi prove in this section. Theore 4.1. Fix τ 2,. When Assuption 1.1ii hods, then there exist rando variabes R τ,a a 1,0], such that as N, where a N = by og og N P H N = 2 ogτ 2 + HN < = PR τ,an = + o1, 4.1 og og N ogτ 2 og og N ogτ 2 1,0]. The distribution of R τ,a, for a 1,0], is given PR τ,a > = P in τ 2 s Y 1 + τ 2 s c Y 2] τ 2 /2 +a Y 1 Y 2 > 0, s Z where c = 1 if is even, and zero otherwise, and Y 1,Y 2 are two independent copies of the iit rando variabe in Outine of the proof We start with an outine of the proof. The proof is divided into severa ey steps proved in 5 subsections, Sections In the first ey step of the proof, in Section 4.2, we spit the probabiity PH N > into separate parts depending on the vaues of Y i,n = τ 2 ogz i,n 1. We prove that PH N >,Y 1,N Y 2,N = 0 = 1 q 2 + o1, N, 4.2 where 1 q is the probabiity that the deayed BP {Z 1 } 1 dies at or before the th generation. When becoes arge, then q q, where q equas the surviva probabiity of {Z 1 } 1. This eaves us to deterine the contribution to PH N > for the cases where Y 1,N further show that for arge enough, and on the event that Y i,n for i = 1,2, where ε > 0 is sa. We denote the event where Y i,n E,N ε, and the event where ax N T ε Y i,n Y i,n Y 2,N > 0. We > 0, whp, Y i,n ε,ε 1 ], ε,ε 1 ], for i = 1,2, by ε for i = 1,2 by F,N ε. The events E,N ε and F,N ε are shown to occur whp, for F,N ε this foows fro Proposition.2a. 720
19 The second ey step in the proof, in Section 4., is to obtain an asyptotic forua for P{H N > } E,N ε. Indeed we prove that for 2 1, and any 1 with 1 1/2, ] P{H N > } E,N ε = E 1 E,N ε F,N εp, 1 + o N,,ε1, 4. where P, 1 is a product of conditiona probabiities of events of the for {H N > H N > 1}. Basicay this foows fro the utipication rue. The identity 4. is estabished in 4.2. In the third ey step, in Section 4.4, we show that, for = N, the ain contribution of the product P, 1 appearing on the right side of 4. is { exp λ N Z 1,N } in 1 +1 Z2,N N 1, B N where λ N = λ N N is in between 1 2 and 4 N, and where B N = B N ε, N defined in 4.51 is such that 1 B N ε, N precisey when T 1,N ε and N 1 T 2,N ε. Thus, by Proposition.2, it ipies that whp 1 +1 N 1 ε 4 Z 1,N τ 1 and Z 2,N N 1 N 1 ε 4 In turn, these bounds aow us to use Proposition.2a. Cobining 4. and 4.4, we estabish in Coroary 4.10, that for a and with og og N N = 2 +, 4.5 ogτ 2 we have { P{H N > N } E,N ε = E 1 E,N ε F,N ε exp λ N { = E 1 E,N ε F,N ε exp λ N where κ = τ 2 1 > 1. τ 1. Z 1,N }] in 1 +1 Z2,N N 1 + o N,,ε1 1 B N exp{κ 1+1 Y 1,N in κ N 1 Y 2,N N 1 }}] + o N,,ε1, 1 B N In the fina ey step, in Sections 4.5 and 4.6, the iniu occurring in 4.6, with the approxiations Y 1,N 1 +1 Y 1,N and Y 2,N N 1 Y 2,N, is anayzed. The ain idea in this anaysis is as foows. With the above approxiations, the right side of 4.6 can be rewritten as { ]}] E 1 E,N ε F,N ε exp λ N exp in κ 1+1 Y 1,N + κ N 1 Y 2,N og + o N,,ε1. 1 B N 4.7 The iniu appearing in the exponent of 4.7 is then rewritten see 4.7 and 4.75 as κ N/2 { in t Z κt Y 1,N + κ c t Y 2,N κ N/2 } og. Since κ N/2, the atter expression ony contributes to 4.7 when in t Z κt Y 1,N + κ c t Y 2,N κ N/2 og
20 Here it wi becoe apparent that the bounds 1 2 λ N 4 are sufficient. The expectation of the indicator of this event eads to the probabiity P in t Z κt Y 1 + κ c t Y 2 κ a N /2,Y 1 Y 2 > 0, with a N and c as defined in Theore 4.1. We copete the proof by showing that conditioning on the event that 1 and 2 are connected is asyptoticay equivaent to conditioning on Y 1 Y 2 > 0. Rear 4.2. In the course of the proof, we wi see that it is not necessary that the degrees of the nodes are i.i.d. In fact, in the proof beow, we need that Propositions.1.4 are vaid, as we as that is concentrated around its ean µn. In Rear A.1.5 in the appendix, we wi investigate what is needed in the proof of Propositions.1.4. In particuar, the proof appies aso to soe instances of the configuration ode where the nuber of nodes with degree is deterinistic for each, when we investigate the distance between two unifory chosen nodes. We now go through the detais of the proof. 4.2 A priory bounds on Y i,n We wish to copute the probabiity PH N >. To do so, we spit PH N > as PH N > = PH N >,Y 1,N Y 2,N = 0 + PH N >,Y 1,N Y 2,N > We wi now prove two eas, and use these to copute the first ter in the righthand side of 4.8. Lea 4.. For any fixed, i N 1,N PY Y 2,N = 0 = 1 q, 2 where q = PY 1 > 0. Proof. The proof is iediate fro Proposition.1 and the independence of Y 1 and Y 2. The foowing ea shows that the probabiity that H N converges to zero for any fixed : Lea 4.4. For any fixed, i PH N = 0. N Proof. As observed above Theore 1.2, by exchangeabiity of the nodes {1,2,...,N}, PH N = P H N, 4.9 where H N is the hopcount between node 1 and a unifory chosen node unequa to 1. We spit, for any 0 < δ < 1, P H N = P H N, Z 1,N N δ + P H N, Z 1,N > N δ
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